#### Task and Configuration space

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We learn the concepts of a robot’s task space and its configuration space, and the relationship between the dimensions of these two spaces.

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We learn the concepts of a robot’s task space and its configuration space, and the relationship between the dimensions of these two spaces.

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We can describe the relationship between a 3D world point and a 2D image plane point, both expressed in homogeneous coordinates, using a linear transformation – a 3×4 matrix. Then we can extend this to account for an image plane which is a regular grid of discrete pixels.

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We learn to compute a trajectory that involves simultaneous smooth motion of many robot joints.

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What are the consequences of representing a three-dimensional scene using only two-dimensions? The appearance of parallel lines converging and circular objects being elliptical should be surprising but we take this for granted.

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Let’s look at how light rays reflected from an object can form an image. We use the simple geometry of a pinhole camera to describe how points in a three-dimensional scene are projected on to a two-dimensional image plane.

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We live in a three-dimensional world but it’s taken humans a long time to learn how to realistically depict the illusion of depth on a flat surface.

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We learn how to describe the position and orientation of objects in the 3-dimensional space that we live in. This builds on our understanding of describing position and orientation in two dimensions.

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We learn how to describe the position and orientation of objects on a 2-dimensional plane. We introduce the notion of reference frames as a basis for describing the position of objects in two dimensions.

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Let’s learn how to import a color image into MATLAB and see how the data is organized as a matrix with three dimensions.

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The Jacobian matrix provides powerful diagnostics about how well the robot’s configuration is suited to the task. Wrist singularities can be easily detected and the concept of a velocity ellipse is extended to a 3-dimensional velocity ellipsoid.